where in the inequality, we use the fact that \(_T^i\)  and \(_T^{\prime i}\)  are Gaussians with means separated by \(2G\) , and variance being \(T\sigma ^2\)  and use the lower bound result on TV between high-dimensional Gaussians [1]}. Finally, in the last equality, we compute the Expected value of \(i\)  under the mixture distribution - recall that \(i\)  is a sum of \(T\)  Bernoulli random variables with bias \(\frac{m}{n}\) , the expectation of which is \(\frac{Tm}{n}\) .
